Finite Group Databasebeta

← #6484474

Subgroups — #6484474

Order 512.

1446 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#36635C211yesmaximal, characteristic
3#33217C214yesmaximal
4#30294C214yesmaximal
5#29318C214yesmaximal, complemented
6#29317C214yesmaximal, complemented
7#29214C212yesmaximal, complemented
8#29211C212yesmaximal, complemented
9#28695C212yesmaximal, complemented
10#28692C212yesmaximal, complemented
11#21960C214yesmaximal
12C2×#1767C211yesmaximal, characteristic
13C2×#501C2214yes—
14C22×C4⋊C8C2212yes—
15C22×#97C2212yes—
16#2864C2212yes—
17#2864C2212yes—
18#2863—22no—
19#2863—22no—
20#2862C2211yescharacteristic
21#2862C2211yescharacteristic
22#2859C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2857C2212yes—
26#2855—22no—
27#2855—22no—
28#2851C2212yes—
29#2851C2212yes—
30C4×C4⋊Q8C2212yes—
31C4×C4⋊Q8C2212yes—
32C4×#125C2212yes—
33C4×#125C2212yes—
34C4×C22⋊Q8—22no—
35C4×C22⋊Q8—22no—
36#2295—22no—
37#2295—22no—
38#2283C2212yes—
39#2283C2212yes—
40C2×C43C2211yescharacteristic
41#2085C2218yes—
42#2084C2214yes—
43#2083C2214yes—
44#2059C2218yes—
45#2058C2214yes—
46#2057C2214yes—
47#1993C2214yes—
48#1992C2214yes—
49#1949C2218yes—
50#1949C2218yes—
51#1949C2218yes—
52#1949C2218yes—
53#1944C2218yes—
54#1943C2218yes—
55#1847C2214yes—
56#1808C2218yes—
57#1807C2218yes—
58#1767C2212yes—
59#1767C2212yes—
60#1767C2212yes—
61#1767C2212yes—
62C2×#336C2212yes—
63C23×C8C2314yes—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×#125C2312yes—
67C2×#125C2312yes—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C22×C42D412yes—
73C22×C42D412yes—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#501—48no—
78C2×#107C2318yes—
79C2×#107C2318yes—
80C2×#106C2318yes—
81C2×#106C2318yes—
82C2×C4⋊C8Q8116yescomplemented
83C2×C4⋊C8C2314yes—
84C2×C4⋊C8C2314yes—
85C2×#97—44no—
86C2×#97C2314yes—
87C2×#97C2314yes—
88#391—22no—
89#391—22no—
90#391—22no—
91#391—22no—
92#391—22no—
93#391—22no—
94#391—22no—
95#391—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389C2314yes—
105#389C2314yes—
106#389C2314yes—
107#389C2314yes—
108#387—24no—
109#387—24no—
110#387—24no—
111#387—24no—
112#387—24no—
113#387—24no—
114#387—24no—
115#387—24no—
116#385—24no—
117#385—24no—
118#384C2312yes—
119#384C2312yes—
120#383—22no—
121#383—22no—
122#382C2312yes—
123#382C2312yes—
124#382C2312yes—
125#382C2312yes—
126#382C2312yes—
127#382C2312yes—
128#381—22no—
129#381—22no—
130#380—22no—
131#380—22no—
132#380C2314yes—
133#380C2314yes—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4C2314yes—
139C4×C4⋊C4C2314yes—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142#376—22no—
143#376—22no—
144C43D412yes—
145C43D412yes—
146C43C2311yescharacteristic
147C43C2311yescharacteristic
148C43C2311yescharacteristic
149C43C2311yescharacteristic
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#336C2314yes—
155#336C2314yes—
156C22×Q8—22no—
157C22×Q8—22no—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8—44no—
164C22×C8—44no—
165C22×C8C2×Q8116yes—
166C22×C8C2418yes—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—44no—
170#125—24no—
171#125—24no—
172#125—44no—
173C22⋊Q8—48no—
174C22⋊Q8—48no—
175C4×Q8—48no—
176C4×Q8—48no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C2414yes—
190C2×C4⋊C4C2414yes—
191C2×C4⋊C4C2414yes—
192C2×C4⋊C4C2414yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—44no—
200C2×C42—42no—
201C2×C42—44no—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×Q818yes—
215C2×C42C2411yescharacteristic
216C2×C42C2411yescharacteristic
217C2×C42C2411yescharacteristic
218C2×C42C2411yescharacteristic
219C2×C42C2411yescharacteristic
220C2×C42C2411yescharacteristic
221#107—48no—
222#107—48no—
223#106—48no—
224#106—48no—
225C4⋊C8C2×Q8116yes—
226C4⋊C8C2×Q8116yes—
227#97—28no—
228#97—44no—
229#95—24no—
230#95—24no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247C24C22×D411yescharacteristic
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—44no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4C4⋊Q814yes—
270C22×C4C4⋊Q814yes—
271C22×C4C22×D412yes—
272C22×C4C22×D412yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D412yes—
279C22×C4C22×D412yes—
280C22×C4C22×Q814yes—
281C22×C4C2511yescharacteristic
282C2×C8—28no—
283C2×C8—28no—
284C2×C8—416no—
285C2×C8C22×Q8116yes—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—28no—
292C4⋊C4—44no—
293C4⋊C4—48no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—48no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—44no—
300C4⋊C4—48no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—48no—
304C22⋊C4—48no—
305C22⋊C4—48no—
306C42—24no—
307C42—24no—
308C42—24no—
309C42—21no—
310C42—21no—
311C42—22no—
312C42—22no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—48no—
321C42—48no—
322C42C4⋊Q814yes—
323C42C4⋊Q814yes—
324C42C22×Q814yes—
325C42C22×Q814yes—
326C42H(5,𝔽2)12yes—
327C422−1+412yes—
328C23—21no—
329C23—21no—
330C23—22no—
331C23C2×C4⋊Q812yes—
332C23#57514yes—
333C23C23×D411yescharacteristic
334Q8—88no—
335Q8—88no—
336C2×C4—28no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4C2×C4⋊Q812yes—
371C2×C4C2×C4⋊Q812yes—
372C2×C4C2×C4⋊Q812yes—
373C2×C4C2×C4⋊Q812yes—
374C2×C4C2×C4⋊Q812yes—
375C2×C4#53714yes—
376C2×C4#57514yes—
377C2×C4C23×Q812yes—
378C2×C4C2×H(5,𝔽2)11yescharacteristic
379C2×C4C2×2−1+411yescharacteristic
380C2×C4#58512yes—
381C8—216no—
382C22—24no—
383C22—22no—
384C22—24no—
385C22—22no—
386C22#316214yes—
387C22C22×C4⋊Q811yescharacteristic
388C22C2×#53711yescharacteristic
389C22#344911yescharacteristic
390C22C2×#57512yes—
391C22C2×#57512yes—
392C4—24no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—24no—
397C4—22no—
398C4—22no—
399C4—42no—
400C4—42no—
401C4—84no—
402C4—84no—
403C4—84no—
404C4—84no—
405C4#350112yes—
406C4#350212yes—
407C2—24no—
408C2#3517311yescharacteristic
409C2C2×#316211yescharacteristic
410C2#6066311yescharacteristic
411C2#6091712yes—
412C2#6095912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #29318C2semidirect
6. #29317C2semidirect
7. #29214C2semidirect
8. #29211C2semidirect
9. #28695C2semidirect
10. #28692C2semidirect
82. C2×C4⋊C8Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#366352
#332172
#302942
#293182
#293172
#292142
#292112
#286952
#286922
#219602
C2×#17672